Drone training, how to train without training data?
It's debatable whether neural networks can find a better solution than PID, if your goal is to simply keep the output around a certain reference point PID should do a perfect job pretty much. If you really want to use "intelligent" control with NN you can look into reinforcement learning.
Few interesting papers that directly adress your problem:  

Autonomous helicopter flight via reinforcement learning
An Application of Reinforcement Learning to Aerobatic Helicopter Flight
Autonomous Helicopter Control Using Reinforcement Learning Policy Search Methods  

You can also find more in the references of these papers.